What are examples of how Theseus is the hero in Shakespeare's A Midsummer Night's Dream?
The term hero is often used synonymously with the term protagonist. A protagonist is the main character in the story who undergoes change by resolving the conflict the protagonist faces. While Theseus is certainly not the hero in the story in the sense of being the protagonist since he never changes, Theseus certainly does do a couple of heroic deeds, meaning brave deeds that save the day.
Theseus first strives to be heroic in the opening scene of the play when first faced by Egeus with his request to put his daughter to death should she continue to refuse to marry Demetrius. Theseus knows he is morally obligated to uphold the law permitting Hermia to be executed or exiled should she continue to disobey her father; yet, Theseus also openly questions the soundness of Egeus's demand that Hermia marry Demetrius. We can tell that Theseus questions the soundness of Egeus's demand, because when Lysander asserts that he is just as rich as Demetrius and asserts Demetrius has been unfaithful by pursuing Helena, Theseus acknowledges hearing of Demetrius's unfaithfulness. We see Lysander accuse Demetrius of being unfaithful in the following:
Demetrius, I'll avouch it to his head,
Made love to Nedar's daughter, Helena,
And won her soul; and she, sweet lady, dotes,
Devoutly dotes, dotes in idolatry,
Upon this spotted and inconstant man. (1.1.106-110)
We see Theseus acknowledge he knows about Demetrius's unfaithfulness in the following:
I must confess that I have heard so much,
And with Demetrius thought to have spoke thereof. (111-12)
More importantly, he takes Demetrius and Egeus aside to have a private conversation with them. While we don't know exactly what Theseus is counseling them about, we can assume based on his reaction to the above that he is questioning them on their decision to insist Hermia marry Demetrius. Challenging Egeus on the soundness of his decision concerning his daughter's marriage would be Theseus's first heroic act.
Since Theseus so strongly objects to Egeus's demand Hermia marry Demetrius, by the time we reach act 4, scene 1, Theseus very quickly grants Demetrius permission to marry Helena the moment Theseus learns Demetrius is now rightfully in love with her. In granting Demetrius permission to marry Helena, Theseus is ruling against Egeus's petition to execute the full force of the law upon Hermia. Theseus states his ruling in the following:
Egeus, I will overbear your will;
For in the temple by and by with us
These couples shall eternally be knit. (4.1.175-78)
Theseus's decision to overrule Egeus's petition is a second heroic act, and the act that saves the day for the couples.